Question

Explain why the domains of the trigonometric functions are restricted when finding the inverse trigonometric functions.

Short answer

The domains of trigonometric functions are restricted when finding inverse trigonometric functions to ensure that these functions are one-to-one and onto (bijective). This in turn enables us to derive their inverses.

Step-by-step solution

Understanding Functions and their Inverses

A function takes an input from its domain and gives an output which is the element in its range. An inverse function, on the other hand, inverts the operation performed by the initial function. However, an inverse can only be derived if the original function is both one-to-one and onto.

Defining Trigonometric Functions

Trigonometric functions include sine, cosine, and tangent functions, among others. Each of these functions has a specific domain and range. For instance, the domain for sine and cosine functions includes all real numbers, while the domain of the tangent function is all real numbers excluding odd multiples of \(\pi/2\). The range of sine and cosine functions is between -1 and 1, while the range for tangent is all real numbers.

Restricting Domains and finding Inverses

For us to find the inverse of a trigonometric function, the function must be one-to-one (each element in the domain relates to a unique element in the range) and onto. This necessitates restricting the domains of these functions. For example, the sine function's domain is usually restricted to [-\(\pi/2\), \(\pi/2\)], thus making it a one-to-one function, and allowing its inverse, \(\sin^{-1}(x)\), to exist.